C.Drift-Diffusion Conduction Equilibrium (.=J=0) j,=0=p,u,E-D,Vp,=-p,μ，VΦ-D,Vp, j.=0=-p.u.E-D.Vp.=p.H.Vo-D.Vp. vo=-D.-Vp.--kIv(lnp.) P+μ q vo-D._Vp.=5 _kTv(Ine-) pu q P,=P。e-qD/kT Boltzmann Distributions P.=-P。etqo/k灯 P.(=0)=-p_(=0)=Po [Potential is zero when system is charge neutral] v---_(.+p.)--P.[e-w/-e/]-2p.sinh 8 8 'kT (Poisson-Boltzmann Equation) Small Potential Approximation: qΦ <1 sinhΦ、qΦ kTkT 20-2p90=0 skT Φ-Φ=0；Ld=1 εkT Debye Length Y2P。q 6.641,Electromagnetic Fields,Forces,and Motion Lecture 7 Prof.Markus Zahn Page 11 of 276.641, Electromagnetic Fields, Forces, and Motion Lecture 7 Prof. Markus Zahn Page 11 of 27 C. Drift-Diffusion Conduction Equilibrium (JJ0 + − = = ) J =0= E D = D + ++ + + ++ + + ρ µ − ∇ρ −ρ µ ∇Φ − ∇ρ J =0= E D = D − −− − − −− − − −ρ µ − ∇ρ ρ µ ∇Φ − ∇ρ ( ) D k T = = ln q + + + + + − ∇Φ − ∇ρ ∇ ρ ρ µ ( ) − − − − − ∇Φ ∇ρ ∇ ρ ρ µ D k T = = ln q q / kT = eo − Φ + ρ ρ Boltzmann Distributions q / kT = eo + Φ − ρ −ρ ρ Φ −ρ Φ ρ + − () () =0 = =0 = o [Potential is zero when system is charge neutral] ( ) 2 q / kT q / kT o o 2 q = = = e e = sinh kT + − −Φ +Φ −ρ Φ ρ +ρ −ρ ρ ∇Φ − − ⎡ ⎤ ⎣ ⎦ ε εε ε (Poisson-Boltzmann Equation) Small Potential Approximation: q 1 kT Φ << q q sinh kT kT Φ Φ ≈ ρ ∇ Φ− Φ ε 2 2 q0 = 0 k T 2 2 d d o k T =0 ; L = L 2 q Φ ∇ Φ− ρ ε Debye Length